Finding rejection region for an exact one-sample binomial test.

Question

Can $n=10$ be used? $n=20?$ $n=25?$ What is the appropriate rejection region for the chosen $n,$ and what are the actual error probabilities when this region is used?

Answer 1

Here is a partial outline to get you started:

Let $X$ be the number of leaky faucets observed out of $n.$ The null hypothesis is $H_0: p = .1$ and alternative is $H_a: p > .1.$ 'Not proceed' means $H_0$ is rejected.

We seek $\alpha = P(\text{Rej}\, |\, p=.1) \le .1$ and $\beta(.3) = P(\text{Fail to Rej}\, |\, p=.3) \le .1.$ Try $n = 25$ and see what happens.

Finding rejection region for specified significance level.

If $X \sim \mathsf{Binom}(n=25, p=.1),$ then $P(X \ge 5) = .0980 < 0.1,$ so we use rejection region $\{5, 6, \dots, 25\}.$ (Computations use R statistical software, where dbinom is a binomial PDF and 5:20 is the rejection region.)

sum(dbinom(4:20, 25, .1))
[1] 0.2364086
sum(dbinom(5:20, 25, .1))
[1] 0.09799362

Finding Type II error for a specified rejection region.

If $X \sim \mathsf{Binom}(n = 25, p=.3),$ then $P(X \le 4) = .0904 < 0.1.$ So using $n = 25$ and rejection region $\{5, 6, \dots, 25\}$ seems to work.

sum(dbinom(0:4, 25, .3))
[1] 0.09047192

It seems that the first image in your Question is an answer to this problem or a similar one (the disagreement in question numbers notwithstanding). I will leave it to you to explain why the other values $n = 10$ and $n = 20$ won't work, and why for $n = 25$ we need to use rejection region $\{5, 6, \dots, 25\}.$

---

In the figure below, blue bars represent the PDF of $\mathsf{Binom}(25,.1)$ and red bars represent the PDF of $\mathsf{Binom}(25,.3).$ The rejection region is to the right of the vertical dotted line.